Here's the question you clicked on:
Cynosure-EPR
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let b= (1,1,1) let n be a scalar we can say b= nL + e where e is a vector orthogonal to L |dw:1328148406952:dw|
use L^T(b - nL)= 0 (L^T means L transpose. e and L perpendicular) to find the scalar n n = L^T b/(L^T L) to find the reflection of b = nL + e, go in the other direction b reflected = nL - e
e= b - nL so b_reflected = nL - b + nL = 2nL -b
Lemme give it a go
Can you explain L^T? in attempting to find n?
the T means transpose. L^T L is just L dot L, and L^T b is L dot b (dot product)
Okay.. so I'm extremely close. I've gotten (-1/9) of: [11] [ 1 ] [11] The book has the same, but positive. I'm guessing my math is just off somewhere.
I get n = L dot b / (L dot L) = 5/9 and e = b - nL = (1,1,1) - (5/9) (2,1,2) = (-1, 4, -1)/9 b_reflected = nL - e = (5/9) (2,1,2) - (-1, 4, -1)/9 = (10,5,10)/9 + (1, -4, 1)/9 b_reflected = (11,1,11)/9 Here's a picture http://www.wolframalpha.com/input/?i=plot+vector+%281%2C1%2C1%29+%2C+vector+%281%2C2%2C1%29%2C++vector%2811%2F9%2C1%2F9%2C11%2F9%29%2C+vector+%28-11%2F9%2C-1%2F9%2C-11%2F9%29
Aha! You are a Godsend. <3